| L(s) = 1 | + 2-s − 7-s − 8-s + 4·13-s − 14-s − 16-s + 12·17-s + 4·19-s + 5·25-s + 4·26-s + 6·29-s + 4·31-s + 12·34-s + 4·37-s + 4·38-s − 6·41-s − 8·43-s + 12·47-s + 5·50-s + 12·53-s + 56-s + 6·58-s + 6·59-s − 8·61-s + 4·62-s + 64-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.377·7-s − 0.353·8-s + 1.10·13-s − 0.267·14-s − 1/4·16-s + 2.91·17-s + 0.917·19-s + 25-s + 0.784·26-s + 1.11·29-s + 0.718·31-s + 2.05·34-s + 0.657·37-s + 0.648·38-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 0.707·50-s + 1.64·53-s + 0.133·56-s + 0.787·58-s + 0.781·59-s − 1.02·61-s + 0.508·62-s + 1/8·64-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.925715946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.925715946\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.965363900033757056542797857688, −9.896832065968700209378951010560, −8.981262373361538721293537345588, −8.974194983815212186385065008631, −8.294617437855192102100987822451, −8.031293774802245118875663112020, −7.48655076274356680955745994755, −7.13726025543389462193913449045, −6.51934795054915739147487627171, −6.20692714034726877009450547907, −5.63891753560402551551713279035, −5.40485379215751779468256095872, −4.95184129186987004969716310671, −4.38023679322460222737079841298, −3.67027232218535153757676445394, −3.45173581884622919931806138410, −3.01118174722901517185218105187, −2.43237795903141877385950354742, −1.10167326553198431223830906085, −1.06526607354851973657984272848,
1.06526607354851973657984272848, 1.10167326553198431223830906085, 2.43237795903141877385950354742, 3.01118174722901517185218105187, 3.45173581884622919931806138410, 3.67027232218535153757676445394, 4.38023679322460222737079841298, 4.95184129186987004969716310671, 5.40485379215751779468256095872, 5.63891753560402551551713279035, 6.20692714034726877009450547907, 6.51934795054915739147487627171, 7.13726025543389462193913449045, 7.48655076274356680955745994755, 8.031293774802245118875663112020, 8.294617437855192102100987822451, 8.974194983815212186385065008631, 8.981262373361538721293537345588, 9.896832065968700209378951010560, 9.965363900033757056542797857688
